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Sci Rep. 2016; 6: 28850.
Published online 2016 June 28. doi:  10.1038/srep28850
PMCID: PMC4924101

Improved Separability Criteria Based on Bloch Representation of Density Matrices

Abstract

The correlation matrices or tensors in the Bloch representation of density matrices are encoded with entanglement properties. In this paper, based on the Bloch representation of density matrices, we give some new separability criteria for bipartite and multipartite quantum states. Theoretical analysis and some examples show that the proposed criteria can be more efficient than the previous related criteria.

Quantum entanglement is a fascinating phenomenon in quantum physics. It can be seen as a physical resource like energy with applications from quantum teleportation to quantum cryptography1,2,3,4,5. In the last years, much work has been devoted to understanding entanglement, but there are still many problems unsolved. One of them is to determine whether a given quantum state is entangled or separable. This problem is extremely difficult to solve, and has been proved as a nondeterministic polynomial-time hard problem6. Nevertheless, a variety of operational criteria for separability of quantum states have been proposed in the last decades. Among them are the positive partial transpose (PPT) criterion or Peres-Horodecki criterion7,8, realignment criteria9,10,11,12,13, covariance matrix criteria14,15,16 and so on; see, e.g.17,18, for a comprehensive survey.

The Bloch representation19,20,21 of density matrices stands as an important role in quantum information. The correlation matrices or tensors in the Bloch representation are encoded with entanglement properties22,23, which can be exploited to study quantum entanglement. In ref. 24, by making use of correlation matrices, Vicente obtained the correlation matrix criterion for bipartite quantum states, which can be more efficient than the PPT criterion7,8 and the computable cross norm or realignment (CCNR) criterion9,10 in many different situations. After that, this criterion was used to give the analytical lower bounds for the entanglement measures: concurrence and tangle25,26, which are good supplement to the lower bounds based on PPT and CCNR criteria. By the matricizations of tensors, the correlation matrix criterion was generalized to detect non-full-separability of multipartite states27. Later, this multipartite criterion was extended and improved to be a much more general case28. Meanwhile, by the standard tensor norm and the norms of matricizations of tensors, some genuine entanglement conditions were derived. In refs 22, 23, some simple geometrical methods based on correlation tensors were presented to detect various multipartite entanglement. By bounding tensor norms for partially separable states and states of limited dimension, Klöckl and Huber29 studied the detection of multipartite entanglement in an experimentally feasible way. In many cases, only few definite measurements are needed. Recently, Li et al.30 presented some separability criteria under the combination of correlation matrices and the Bloch vectors of reduced density matrices, which can be stronger than the correlation matrix criterion24 by examples.

This paper is further devoted to an investigation of entanglement detection in terms of Bloch representations of density matrices. On the one hand, by adding some parameters, a more general separability criterion for bipartite states is presented, which can outperform the corresponding criteria given in24,30. On the another hand, the presented bipartite separability criterion is extended to the multipartite case. An example shows that the new multipartite separability criterion can be better than the corresponding criteria obtained in refs 27, 28 and 30.

Results

Separability criteria for bipartite states

Let An external file that holds a picture, illustration, etc.
Object name is srep28850-m1.jpg be the traceless Hermitian generators of SU(d) satisfying the orthogonality relation An external file that holds a picture, illustration, etc.
Object name is srep28850-m2.jpg. Then any state ρ in An external file that holds a picture, illustration, etc.
Object name is srep28850-m3.jpg can be represented as21

An external file that holds a picture, illustration, etc.
Object name is srep28850-m4.jpg

where Id denotes the d × d identity matrix,

An external file that holds a picture, illustration, etc.
Object name is srep28850-m5.jpg

Denote by ||·||tr, ||·||2 and Ep×q the trace norm (the sum of singular values), the spectral norm (the maximum singular value) and the p × q matrix with all entries being 1, respectively. By defining An external file that holds a picture, illustration, etc.
Object name is srep28850-m6.jpg, An external file that holds a picture, illustration, etc.
Object name is srep28850-m7.jpg and T = (tij), we construct the following matrix

An external file that holds a picture, illustration, etc.
Object name is srep28850-m8.jpg

where α and β are nonnegative real numbers, m is a given natural number, t stands for transpose, and for any column vector x,

An external file that holds a picture, illustration, etc.
Object name is srep28850-m9.jpg

Using An external file that holds a picture, illustration, etc.
Object name is srep28850-m10.jpg, we can get the following separability criterion for bipartite states.

Theorem 1. If the state ρ in An external file that holds a picture, illustration, etc.
Object name is srep28850-m11.jpg is separable, then

An external file that holds a picture, illustration, etc.
Object name is srep28850-m12.jpg

See Methods for the proof of Theorem 1.

When α and β are chosen to be 0, Theorem 1 reduces to the correlation matrix criterion in ref. 24: if ρ in An external file that holds a picture, illustration, etc.
Object name is srep28850-m13.jpg is separable, then

An external file that holds a picture, illustration, etc.
Object name is srep28850-m14.jpg

If we choose α = β = m = 1, then Theorem 1 becomes the separability criterion given in [30, Corollary 2]: any separable state ρ in An external file that holds a picture, illustration, etc.
Object name is srep28850-m15.jpg must satisfy

An external file that holds a picture, illustration, etc.
Object name is srep28850-m16.jpg

For simplicity, we call these criteria in (6) and (7) the V-B and L-B criteria, respectively. The following result can help us find that our separability criterion from Theorem 1 is stronger than the V-B and L-B criteria.

Proposition 1. If α and β are selected to satisfy

An external file that holds a picture, illustration, etc.
Object name is srep28850-m17.jpg

then Theorem 1 becomes more effective when m gets larger.

See Methods for the proof of Proposition 1.

From Proposition 1, Theorem 1 with the condition (8) is stronger than the V-B criterion.

For the case d1 = d2 and α = β, it follows from Proposition 1 that Theorem 1 is more efficient when m gets larger. In particular, Theorem 1 is better than the L-B criterion, and the L-B criterion is better than the V-B criterion. For the case d1  d2, let us consider the following example. The following 2 × 4 bound entangled state is due to31:

An external file that holds a picture, illustration, etc.
Object name is srep28850-m18.jpg

where 0 < b < 1. To verify the efficiency of the present criteria, we consider the state

An external file that holds a picture, illustration, etc.
Object name is srep28850-m19.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep28850-m20.jpg. For simplicity, we choose

An external file that holds a picture, illustration, etc.
Object name is srep28850-m21.jpg

Then Theorem 1 can detect the entanglement in ρx for 0.2235  x  1, while the V-B criterion and L-B criterion can only detect the entanglement in ρx for 0.2293  x  1 and 0.2841  x  1, respectively. Thus, Theorem 1 is better than the V-B and L-B criteria.

Separability criteria for multipartite states

Let An external file that holds a picture, illustration, etc.
Object name is srep28850-m22.jpg be an f1 × ··· × fN tensor, A and An external file that holds a picture, illustration, etc.
Object name is srep28850-m23.jpg be two nonempty subsets of {1, ···, N} satisfying An external file that holds a picture, illustration, etc.
Object name is srep28850-m24.jpg. Then we denote by An external file that holds a picture, illustration, etc.
Object name is srep28850-m25.jpg the An external file that holds a picture, illustration, etc.
Object name is srep28850-m26.jpg matricization of An external file that holds a picture, illustration, etc.
Object name is srep28850-m27.jpg; see28 for detail. This matricization is a generalization of mode-n matricization in the multilinear algebra32.

For any state ρ in An external file that holds a picture, illustration, etc.
Object name is srep28850-m28.jpg, we import a natural number m and nonnegative real parameters α1, ···, αN, and define

An external file that holds a picture, illustration, etc.
Object name is srep28850-m29.jpg

We define the tensor An external file that holds a picture, illustration, etc.
Object name is srep28850-m30.jpg with elements

An external file that holds a picture, illustration, etc.
Object name is srep28850-m31.jpg

Clearly, if m = 0, the tensor An external file that holds a picture, illustration, etc.
Object name is srep28850-m32.jpg reduces to the correlation tensor in ref. 27. When m = α1 = ··· = αN = 1, the tensor An external file that holds a picture, illustration, etc.
Object name is srep28850-m33.jpg becomes the tensor with a constant multiple in ref. 30.

An n partite sate ρ in An external file that holds a picture, illustration, etc.
Object name is srep28850-m34.jpg is (fully) separable33 if and only if it can be written in the form

An external file that holds a picture, illustration, etc.
Object name is srep28850-m35.jpg

where the probabilities An external file that holds a picture, illustration, etc.
Object name is srep28850-m36.jpg, and An external file that holds a picture, illustration, etc.
Object name is srep28850-m37.jpg are pure states of the subsystems.

In the following, we give the full separability criterion based on An external file that holds a picture, illustration, etc.
Object name is srep28850-m38.jpg.

Theorem 2. If the state ρ in An external file that holds a picture, illustration, etc.
Object name is srep28850-m39.jpg is fully separable, then, for any subset A of {1, ···, N}, we have

An external file that holds a picture, illustration, etc.
Object name is srep28850-m40.jpg

See Methods for the proof of Theorem 2.

For the case α1 = ··· = αN = 0, Theorem 2 reduces to the criterion given in [28, Theorem 4], which has an important improvement on the corresponding criterion given in ref. 27. If α1 = ··· = αN = 1 and m = 1, then Theorem 2 becomes [30, Corollary 3]. For simplicity, we call these criteria in refs 27, 28 and 30 V-M, H-M and L-M criteria, respectively. In the following we give a tripartite example to demonstrate the efficiency of Theorem 2. Consider a perturbation of the tripartite GHZ state16:

An external file that holds a picture, illustration, etc.
Object name is srep28850-m41.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep28850-m42.jpg is a given real parameter, and γ denotes the normalization. We consider the mixture of this state with the maximally mixed state:

An external file that holds a picture, illustration, etc.
Object name is srep28850-m43.jpg

In the tripartite case, the V-M criterion is equivalent to the H-M criterion obviously. By taking m = 1 and α1 = α2 = α3 = 0.1, Table 1 displays the detection results with different values of An external file that holds a picture, illustration, etc.
Object name is srep28850-m44.jpg. Clearly, Theorem 2 is more efficient than the V-M, H-M and L-M criteria.

Table 1
Entanglement conditions of An external file that holds a picture, illustration, etc.
Object name is srep28850-m67.jpg with different values of An external file that holds a picture, illustration, etc.
Object name is srep28850-m68.jpg from the V-M (H-M) criterion, the L-M criterion and Theorem 2 with α1 = α2 = α3 = 0.1 and m = 1.

Discussions

Correlation matrices or tensors in the Bloch representation of quantum states contain the information of entanglement of the quantum states. Based on the Bloch representation of quantum states, we have given some new separability criteria including the V-B, L-B, V-M, H-M and L-M criteria as special cases. For bipartite cases, by choosing some special parameters involved, our criteria are stronger than the V-B and L-B criteria. For multipartite cases, by a simple example it has been also shown that our criterion can be more efficient than the V-M, H-M and L-M criteria.

Nevertheless, the problem of how to choose the involved parameters such that Theorems 1–2 can detect more entangled states needs to be further studied in the future. In the Bloch representation (1), the traceless Hermitian generators of SU(d) come from Gell-Mann matrices. But this is by far not the only possible choice. Maybe the new basis of observables34 constructed from Heisenberg-Weyl operators can be used to obtain better separable criteria, since the Heisenberg-Weyl based observables can outperform the canonical basis of generalized Gell-Mann operators in entanglement detection34. Thus, this problem is worth studying in the coming days.

It should be noted that the separability criteria Theorems 1–2 presented in30 for bipartite and multipartite states are at most as good as the corresponding V-B, L-B, V-M and L-M criteria, respectively. For example, set An external file that holds a picture, illustration, etc.
Object name is srep28850-m45.jpg. It was shown by [30, Theorem 1] that any separable state ρ in An external file that holds a picture, illustration, etc.
Object name is srep28850-m46.jpg satisfies

An external file that holds a picture, illustration, etc.
Object name is srep28850-m47.jpg

where M = (mij) is any real An external file that holds a picture, illustration, etc.
Object name is srep28850-m48.jpg matrix. From (18) and35, we get

An external file that holds a picture, illustration, etc.
Object name is srep28850-m49.jpg

which implies that the L-B criterion is at least as good as the criterion (18). Other cases can be proved similarly.

Methods

Proof of Theorem 1. Since ρ is separable, from [24, (17)], it follows that there exist vectors An external file that holds a picture, illustration, etc.
Object name is srep28850-m50.jpg and An external file that holds a picture, illustration, etc.
Object name is srep28850-m51.jpg such that

An external file that holds a picture, illustration, etc.
Object name is srep28850-m52.jpg

where

An external file that holds a picture, illustration, etc.
Object name is srep28850-m53.jpg

Thus, the matrix An external file that holds a picture, illustration, etc.
Object name is srep28850-m54.jpg can be written as

An external file that holds a picture, illustration, etc.
Object name is srep28850-m55.jpg

and then

An external file that holds a picture, illustration, etc.
Object name is srep28850-m56.jpg

where we have used the following equality, for any vectors |aright angle bracket and |bright angle bracket,

An external file that holds a picture, illustration, etc.
Object name is srep28850-m57.jpg

Proof of Proposition 1. For any state ρ, from [24, Lemma 1], we get

An external file that holds a picture, illustration, etc.
Object name is srep28850-m58.jpg

If the inequality from (5),

An external file that holds a picture, illustration, etc.
Object name is srep28850-m59.jpg

holds, then from (25) we have

An external file that holds a picture, illustration, etc.
Object name is srep28850-m60.jpg

where the equality (24) has been used in the first and fifth equalities, and, in the third and fourth equalities, we have employed the fact that the trace norm of a Hermitian positive semidefinite matrix is equal to its trace.

Proof of Theorem 2. Without loss of generality, we assume that

An external file that holds a picture, illustration, etc.
Object name is srep28850-m61.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28850-m62.jpg

Since ρ is fully separable, then from27 there exist vectors An external file that holds a picture, illustration, etc.
Object name is srep28850-m63.jpg such that

An external file that holds a picture, illustration, etc.
Object name is srep28850-m64.jpg

where

An external file that holds a picture, illustration, etc.
Object name is srep28850-m65.jpg

Thus,

An external file that holds a picture, illustration, etc.
Object name is srep28850-m66.jpg

where we have used the equality (24).

Additional Information

How to cite this article: Shen, S.-Q. et al. Improved Separability Criteria Based on Bloch Representation of Density Matrices. Sci. Rep. 6, 28850; doi: 10.1038/srep28850 (2016).

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (No. 15CX05062A, No. 15CX02075A, No. 16CX02049A), Qingdao Applied Basic Research Program (No. 15-9-1-103-jch), NSFC (No. 11275131) and the Project-sponsored by SRF for ROCS, SEM. We are grateful to the referee for their helpful suggestions to improve the quality of this paper.

Footnotes

Author Contributions S.-Q.S., M.L. and S.-M.F. wrote the main manuscript text. J.Y. computed the examples. All authors reviewed the manuscript.

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